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Theorem brabga 4498
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brabga.2  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabga  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)    V( x, y)    W( x, y)

Proof of Theorem brabga
StepHypRef Expression
1 df-br 4238 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabga.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2506 . . 3  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
41, 3bitri 242 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
5 opelopabga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
65opelopabga 4497 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
74, 6syl5bb 250 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   <.cop 3841   class class class wbr 4237   {copab 4290
This theorem is referenced by:  braba  4501  brabg  4503  epelg  4524  brcog  5068  fmptco  5930  ofrfval  6342  wemaplem1  7544  oemapval  7668  wemapwe  7683  fpwwe2lem2  8538  fpwwelem  8551  clim  12319  rlim  12320  vdwmc  13377  isstruct2  13509  brssc  14045  isfunc  14092  isfull  14138  isfth  14142  ipole  14615  eqgval  15020  frgpuplem  15435  dvdsr  15782  ulmval  20327  isuhgra  21369  isumgra  21381  isuslgra  21403  isusgra  21404  isausgra  21410  iscusgra  21496  iswlkon  21562  istrlon  21572  ispthon  21607  isspthon  21614  isconngra  21690  isconngra1  21691  iseupa  21718  hlimi  22721  fmptcof2  24107  isinftm  24282  metidv  24318  brae  24623  braew  24624  brfae  24630  islindf  27297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292
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